What value is used to construct the confidence interval when the standard deviation is unknown?

When the standard deviation of a population is unknown, the appropriate value to construct the confidence interval is the t-value. The t-distribution is specifically designed for situations where the sample size is small (typically less than 30) or when the population standard deviation is not known, which is common in practical applications.

The t-value accounts for the added uncertainty due to the estimation of the population standard deviation from the sample. As the sample size increases, the t-distribution approaches the normal distribution, and thus the t-value and z-value become increasingly similar. However, for smaller samples or when the standard deviation is unknown, using the t-value provides a more accurate estimation of the margin of error and subsequently, the confidence interval.

In contrast, the z-value is used when the population standard deviation is known and is typically applicable to larger sample sizes where normality can be assumed. The mean value itself is not a metric used for constructing the confidence interval; it is simply the central tendency of the sample data. The margin of error is calculated using the t-value (or z-value) alongside the standard error but is not itself a value on which the interval is constructed.

Thus, the use of the t-value facilitates a more reliable estimation for confidence intervals under